Pointwise Operation/Examples

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Examples of Pointwise Operations

Cube and Sine Functions

Let $f$ and $g$ be the real functions defined as

\(\ds \forall x \in \R: \, \) \(\ds \map f x\) \(=\) \(\ds x^3\)
\(\ds \forall x \in \R: \, \) \(\ds \map g x\) \(=\) \(\ds \sin x\)


Then for all $x \in \R$:

\(\ds \map {\paren {f + g} } x\) \(=\) \(\ds x^3 + \sin x\)
\(\ds \map {\paren {f - g} } x\) \(=\) \(\ds x^3 - \sin x\)
\(\ds \map {\paren {f \times g} } x = \map {\paren {g \times f} } x\) \(=\) \(\ds x^3 \sin x\)
\(\ds \map {\paren {f \times f} } x\) \(=\) \(\ds x^6\)


Contrast this with:

\(\ds \map {\paren {f \circ g} } x\) \(=\) \(\ds \paren {\sin x}^3\)
\(\ds \map {\paren {g \circ f} } x\) \(=\) \(\ds \map \sin {x^3}\)
\(\ds \map {\paren {f \circ f} } x\) \(=\) \(\ds x^9\)